Stochastic representation for forward and backward general diffusion equations with space-time-dependent coefficients and external forcing

In this paper, we develop a novel and rigorous approach to derive the stochastic representation for the forward and backward general diffusion equations with space-time-dependent coefficients and external forcing, where the general diffusion operator is the infinitesimal generator of some general time-homogeneous Markov process Y on Rd(d≥2) that is in weak duality with another Markov process Y* on Rd. First, after deriving the infinitesimal generators of the stochastic integral equations driven by Y or Y* with time-dependent coefficients, we prove the global existence and uniqueness of the solutions for the forward and backward general diffusion equations based on the Feynman–Kac formula. Notably, the classical method based on Itô’s formula can no longer be used. Then, we establish the stochastic representation for the forward and backward d-dimensional quasi-geostrophic equations and Navier–Stokes equations given in terms of stochastic integral equations driven by the general Markov process Y. Finally, the Markov process Y and its dual Y* are illustrated with intriguing examples including Itô diffusion, Lévy process, symmetric stable Lévy process, and tempered stable Lévy process, and the corresponding infinitesimal generator and dual operator are presented as applications.